A 60 W bulb is placed at a distance of 4 m from you. The bulb is emtting light of wavelength 600 nm uniformly in all directions. In 0.1 s, how many photons enter your eye if the pupil of the eye is having a diameter of 2mm? [take `hc=1240eV-nm]`

Options

A. `2.84xx10^(12)`
B. `2.84xx10^(11)`
C. `9.37xx10^(11)`
D. `6.48xx10^(11)`

Explanation

To solve the problem step by step, we will follow the outlined procedure to find the number of photons entering the eye from a 60 W bulb emitting light of wavelength 600 nm.

### Step 1: Calculate the intensity of light at the distance of the eye
The intensity \( I \) of light emitted uniformly from a point source is given by the formula:

\[ I = \frac{P}{4 \pi R^2} \]

Where:
- \( P = 60 \, \text{W} \) (power of the bulb)
- \( R = 4 \, \text{m} \) (distance from the bulb)

Substituting the values:

\[ I = \frac{60}{4 \pi (4)^2} = \frac{60}{4 \pi \cdot 16} = \frac{60}{64 \pi} = \frac{15}{16 \pi} \, \text{W/m}^2 \]

### Step 2: Calculate the area of the pupil of the eye
The area \( A \) of the pupil can be calculated using the formula for the area of a circle:

\[ A = \pi \left(\frac{D}{2}\right)^2 \]

Where:
- \( D = 2 \, \text{mm} = 2 \times 10^{-3} \, \text{m} \)

Substituting the value:

\[ A = \pi \left(\frac{2 \times 10^{-3}}{2}\right)^2 = \pi \left(10^{-3}\right)^2 = \pi \times 10^{-6} \, \text{m}^2 \]

### Step 3: Calculate the power entering the eye
The power \( P_1 \) entering the eye is given by:

\[ P_1 = I \times A \]

Substituting the values we calculated:

\[ P_1 = \left(\frac{15}{16 \pi}\right) \times \left(\pi \times 10^{-6}\right) = \frac{15 \times 10^{-6}}{16} \, \text{W} \]

Calculating this gives:

\[ P_1 = \frac{15 \times 10^{-6}}{16} = 9.375 \times 10^{-7} \, \text{W} \]

### Step 4: Calculate the energy of a single photon
The energy \( E \) of a single photon can be calculated using the formula:

\[ E = \frac{hc}{\lambda} \]

Where:
- \( h = 1240 \, \text{eV-nm} \)
- \( c = 1.6 \times 10^{-19} \, \text{J/eV} \)
- \( \lambda = 600 \, \text{nm} \)

Substituting the values:

\[ E = \frac{1240 \times 1.6 \times 10^{-19}}{600} = \frac{1984 \times 10^{-19}}{600} \approx 3.3067 \times 10^{-19} \, \text{J} \]

### Step 5: Calculate the number of photons entering the eye per second
The number of photons \( n \) entering the eye per second can be calculated using:

\[ P_1 = n \times E \]

Rearranging gives:

\[ n = \frac{P_1}{E} \]

Substituting the values:

\[ n = \frac{9.375 \times 10^{-7}}{3.3067 \times 10^{-19}} \approx 2.84 \times 10^{12} \, \text{photons/second} \]

### Step 6: Calculate the number of photons in 0.1 seconds
To find the total number of photons entering the eye in 0.1 seconds:

\[ \text{Total photons} = n \times 0.1 = 2.84 \times 10^{12} \times 0.1 = 2.84 \times 10^{11} \, \text{photons} \]

### Final Answer
The total number of photons entering your eye in 0.1 seconds is approximately \( 2.84 \times 10^{11} \) photons.

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